Distance/time Graphs

Distance/time graphs A distance/time graph shows how an objects’ distance changes with time. What do we know about the movement of this object? This object is not moving. Its distance stays the same as time goes by so it is not moving.

Distance/time graphs What do we know about the movement of this object? This object’s distance increases as time goes by. The object is moving and so it has speed.

Distance/time graphs What do we know about the movement of this object? This slope is more steep so the object is moving away quicker. This means that the object is moving quicker.

Distance/time graphs What do we know about the movement of this object? The slope is down rather than up this time. This means that the object is moving in the opposite direction or coming back.

Distance/time graphs

Matching a Graph to a Story A. Tom took his dog for a walk to the park. He set off slowly and then increased his pace. At the park Tom turned around and walked slowly back home. Distance from home Time B. Tom rode his bike east from his home up a steep hill. After a while the slope eased off. At the top he raced down the other side. C. Tom went for a jog. At the end of his road he bumped into a friend and his pace slowed. When Tom left his friend he walked quickly back home.

Matching Cards Take turns at matching pairs of cards (graph and story cards only). You may want to take a graph and find a story that matches it. Alternatively, you may prefer to take a story and find a graph that matches it. Each time you do this, explain your thinking clearly and carefully. If you think there is no suitable card that matches, write one of your own. Place your cards side by side on your large sheet of paper, not on top of one another, so that everyone can see them. Write your reasons for the match paper just as we did with the example in class. Make sure you leave plenty of space around the cards as, eventually, you will be adding another card to each matched pair.

Sharing Work One student from each group is to share their answers. Be ready to explain the reasons for your group's matches.

Graph data Now, in your groups, work out which data table matches each graph and story.

Distance/time graphs Jack woke up and he was very late for school and so he ran as fast as he could (PART A). He then stayed in school for the day (PART B). When school finished Jack walked home very slowly (PART C).

Distance/time graphs Story 1 Dave woke up and he was early for work. He got in his car and drove slowly to work. In work Dave sat in the office for the whole day. At the end of the day Dave drove fast so he could watch the Man U game. Story 2 Polly started the cross country race as fast as she could. After a while she was so tired that she had to stop for a rest. Once she had got her breath back she ran slowly to the finish line.

Distance/time graphs Story 3 It was a bright summer day and Dan decided to go on a bike ride. So he got on his bike and cycled slowly for the whole morning. Dinner time, Dan stopped to have a burger in McDonalds. After lunch Dan decided to go on for another few miles cycling slowly. After a while it got dark and so Dan cycled home quickly. Story 4 Tina was hungry so she decided to go to spar for some crisps. On her way there she stopped at the ice cream shop. After eating her ice cream she carried on walking to spar. When she arrived at spar she was not hungry anymore and so she ran quickly over to her friends house.

Distance/time graphs Story 5 Peter woke up and he was very late for school and so he ran as quickly as he could. He then sat in lessons until dinner. Dinner time Peter ran home quickly because he had forgot his Math's book. After picking up his book he ran back to school quickly before the end of dinner. In the afternoon he sat through a double Math's lesson. At the end of the day Peter walked home slowly.

Distance/time graphs Sunday started a bit cloudy but I thought I’d keep to the original plan and go cycling. What I usually do is to cycle from the bike-hire place to a pub where I can sit outside and have some lunch – burger and chips probably. The pub’s not far – about 5 km from the start and it takes me about 30 minutes to get there. I usually stop for about 45 minutes for lunch. After lunch, I go on another 2 km which takes about 15 minutes. I then head back to the bike-hire place which takes 60 minutes.

Finding speed with the slope of a line The slope (gradient) of a line on a distance/time graph represents the object’s speed. ALWAYS: Draw in a triangle that takes up over half the straight line Calculate it using: Gradient = rise/step or Gradient = change in y change in x

Practice with these on mini white boards 4/2 = 2 2/6 = 0.33 -6/2 = -3 3/3 = 1 -4/2 = 2 -2/6 = -0.33 5/1 = 5 -5/1 = -5 2/10 = 5 -3/4 = -0.75 Answers on click

Measuring speed practical

Measuring speed practical a Use books or wooden blocks to support one end of the runway and make the board slope. The slope should be big enough for the trolley to gently increase in speed as it travels down it. Put a small piece of tape on the runway every 25cm. b Let the trolley start from rest (standstill). Time the trolley, as well as you can, over the first 25 cm it travels. Time it over the second 25 cm, over the third 25 cm, and so on.    c Repeat the measurements twice again. See how much variation there is in your measurements.    d Write all your measurements in a table. Work out the average (mean) time for the first 25 cm, for the second 25 cm, and so on.    e Make a line graph to show the information about distance and time.    f Divide the distance by the time. That will tell you the mean velocity of the trolley during each part of its journey. Make a line graph of this velocity information.    g Note the patterns you see in the changes that take place to the time, the distance, and the velocity.  EXT – investigate changing the slope

Drawing the perfect graph Turns up in papers 2, 3 and 6. Worth up to 5 marks each. That’s 5/40 or 12.5% of paper 6 for just one question. Or more than 1 grade! Worth doing properly?

What to do….. Chose a suitable scale and label the axis correctly. Plot all the points as accurately as you can. Use a sharp pencil taking care to plot each point with a small, neat cross. Draw a thin, best fit line. The y-axis (dependant variable) is the vertical one. The x-axis (independent variable) is the horizontal one. You will be told which quantity to plot on the y-axis and which is on the x-axis.

When plotting a graph it is important to choose the scales so that the plots occupy more than half of the graph grid. Careless, rushed graph plotting can lose several marks. You should understand that if y is proportional to x then the graph will be a straight line through the origin. Graph axis must be labelled with the quantity and unit as in these examples: I/A, or t/s, or y/m.

Common mistakes … And … zig-zag line is inappropriate. What’s wrong? Y axis scales is poor. Why? Cannot divide 3 by 5 easily to find other points on the line What’s better? Go up in 5s And … zig-zag line is inappropriate.

Common mistakes … What’s wrong? Red best fit line poorly judged Why? Doesn’t have roughly equal space each side What’s better? Black line

Common mistakes … Spot 3 errors Thick line near origin Line is no continuous at x = 2.7 The line ignores the last 3 points. What’s better? Draw one thin, smooth best fit line.

Common mistakes … What’s wrong? Y axis scales is poor. Why? Big gap from 0 to 60. What’s better? Use a ‘false’ origin starting at 50 or 60

Now draw your own … Use your data to plot a distance / time graph Look out for any anomalous results (outliers) first, identify and ignore them Choose axes & scales Plot points Going through the origin? (proportional data) Draw best fit line. Score 5/5

Speed and velocity You’ve now calculated the speed of an object moving in a straight line. But what if it goes round a corner? This bike is travelling at a constant speed, but its VELOCITY is changing. Velocity is SPEED in a given DIRECTION.

SCALARS and VECTORS Speed is a SCALAR quantity. It doesn’t tell us the direction, just the magnitude (size) Velocity is the speed in a given direction Velocity is a VECTOR quantity. It tells us the magnitude and direction. (copy this)

If velocity has size and direction, what are they? The size is the speed The direction is … the direction! Consider this: These cars have (roughly) the same speed. They have velocities with opposite directions.

What’s Damien’s speed? What’s his velocity? Watch the clip and describe both his velocity and speed

Changing velocity = acceleration and coming back and going away Concentrate on the yellow and blue lines. On a distance, time graph, a curved line is a changing speed. Copy the yellow and blue lines.

Acceleration On a velocity/time graphs, things look different: Watch, we’ll come back later A straight slope means speeding up or slowing down – that’s a change in velocity, or ACCELERATION. (Sketch this)

Distance travelled The graph tells us more … Consider the area under the graph … Area of a triangle = 1/2xbxh 1/2x10x20 = 100 20x20 = 400 1/2x20x40 = 400 100 + 400 + 400 = 900 But 900 what? m/s x s = just metres So this object travelled 900 m Area under a velocity/time graph tells us the distance travelled.

Working out acceleration Acceleration is the change in velocity in a given time. So if a car accelerates from 0 to 10 m/s in 20 seconds, its acceleration is: (10 minus 0)/20 = 10/20 = 0.5 m/s/s (yes … 0.5 metres per second per second) Have a think about what that actually means: The car’s velocity increased by half a metre per second each second.

Δ is the Greek letter “delta” which stands for change. And there’s a formula: a = acceleration Δv = change in velocity t = time Δ is the Greek letter “delta” which stands for change. Sometimes it’s written:

Back to the Lambourghini … Now while you are watching, time how long it takes for the car to get to top speed. Insert this into the equation: a = (280 – 0) / 19 = 14.74 km/h/s (= 14.74 kmh-2) a = (78 – 0) / 19 = 4.1 m/s/s (= 4.1 ms-2) Just think about how fast that is. Travelling at 78 m/s at the end.

Worked example A Cheetah accelerates from standing to 20 m/s in 4 seconds. What is its acceleration? acceleration = change in velocity / time = (20 – 0) / 4 = 5 m/s/s A car accelerates from 10 m/s to 20 m/s in 3 seconds. What is its acceleration? acceleration = change in velocity / time = (20 – 10) / 4 = 10 / 4 = 2.5 m/s/s

Prep Acceleration worksheet